Transcript Algebraic Model

INNOVATION LECTURES
(I N N O l E C)
Binding and Kinetics for Experimental Biologists
Lecture 1
Numerical Models for Biomolecular Interactions
Petr Kuzmič, Ph.D.
BioKin, Ltd.
WATERTOWN, MASSACHUSETTS, U.S.A.
Lecture outline
•
The problem:
Traditional equations for fitting biomolecular binding data
restrict the experimental design. Typically, at least one component
must be present in very large excess.
•
The solution:
Abandon algebraic equations entirely. Use iterative numerical models,
which can be derived automatically by the computer.
•
An implementation:
Software DynaFit.
•
An example:
Kinetics of forked DNA binding to the protein-protein complex formed
by DNA-polymerase sliding clamp (gp45) and clamp loader (gp44/62).
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Traditional approach is based on algebraic models
A typical “cookbook”
for experimental biologists
molecular mechanism
algebraic model
Cold Spring Harbor Laboratory Press
Cold Spring Harbor, NY, 2007
ISBN-10: 0879697369
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Algebraic models restrict experiment design
Goodrich & Kugel (2006) Binding and Kinetics for Molecular Biologists
EXAMPLE: Determine the association rate constant for A + B  AB
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Experimental handbooks are full of restrictions
Goodrich & Kugel (2007) Binding and Kinetics for Molecular Biologists
p. 34
p. 79
p. 120
p. 126
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Numerical vs. algebraic mathematical models
FROM A VARIETY OF ALGEBRAIC EQUATIONS TO A UNIFORM SYSTEM OF DIFFERENTIAL EQUATIONS
EXAMPLE: Determine the rate constant k1 and k-1 for A + B
k1
AB
k-1
ALGEBRAIC EQUATIONS
DIFFERENTIAL EQUATIONS
d[A]/dt = -k1[A][B] + k-1[AB]
d[B]/dt = -k1[A][B] + k-1[AB]
d[AB]/dt = +k1[A][B] – k-1[AB]
Applies only when [B] >> [A]
Applies under all conditions
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Advantages and disadvantages of numerical models
THERE IS NO SUCH THING AS A FREE LUNCH
ADVANTAGE
ALGEBRAIC
MODEL
DIFFERENTIAL
MODEL
can be derived for any molecular mechanism
-
+
can be derived automatically by computer
-
+
can be applied under any experimental conditions
-
+
can be evaluated without specialized software
+
-
requires very little computation time
+
-
does not require an initial estimate
+
-
is resistant to truncation and round-off errors
+
-
has a long tradition: many papers published
+
-
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Specialized numerical software: DynaFit
MORE THAN 600 PAPERS PUBLISHED WITH IT (1996 – 2009)
2009
DOWNLOAD http://www.biokin.com/dynafit
FREE for academic users
Kuzmic (2009) Meth. Enzymol., 467, 247-280
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DynaFit software: Citation analysis
APPROXIMATELY 850 JOURNAL ARTICLES PUBLISHED SINCE 1998
Kuzmic, P. (1996) "Program DYNAFIT for the analysis of enzyme kinetic data:
Application to HIV proteinase" Anal. Biochem. 237, 260-273.
Kuzmic, P. (2009) "DynaFit - A software package for enzymology“
Meth. Enzymol. 467, 247-280.
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Theoretical foundations: Mass Action Law
RATE IS PROPORTIONAL TO CONCENTRATION(S)
“rate” … “derivative”
MONOMOLECULAR REACTIONS
A
products
rate is proportional to [A]
- d [A] / d t = k [A]
monomolecular rate constant
1 / time
BIMOLECULAR REACTIONS
A+B
products
rate is proportional to [A]  [B]
- d [A] / d t = - d [B] / d t = k [A]  [B]
bimolecular rate constant
1 / (concentration  time)
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Theoretical foundations: Mass Conservation Law
PRODUCTS ARE FORMED WITH THE SAME RATE AS REACTANTS DISAPPEAR
EXAMPLE
A
- d [A] / d t = + d [P] / d t = + d [Q] / d t
P+Q
COMPOSITION RULE
ADDITIVITY OF TERMS FROM SEPARATE REACTIONS
mechanism:
A
B
k1
k2
B
d [B] / d t = + k1 [A] - k2 [B]
C
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Composition Rule: Example
EXAMPLE MECHANISM
k+1
E+ A
EA
RATE EQUATIONS
d[P] / d t = + k+5 [EAB]
k-1
k+2
EAB
EA + B
d[EAB] / d t = + k+2 [EA][B]
k-2
- k-2 [EAB]
k+3
+ k+4 [EB][A]
E+ B
EB
- k-4 [EAB]
k-3
- k+5 [EAB]
k+4
EAB
EB + A
k-4
k+5
EAB
E+ P+ Q
Similarly for other species...
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A "Kinetic Compiler"
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
k1
E.S
E+S
k3
E+P
k2
Rate terms:
Input (plain text file):
E + S ---> ES
:
k1
k1  [E]  [S]
ES ---> E + S
:
k2
k2  [ES]
ES ---> E + P
:
k3
k3  [ES]
Rate equations:
d[E ] / dt = - k1  [E]  [S]
+ k2  [ES]
+ k3  [ES]
d[ES ] / dt = + k1  [E]  [S]
- k2  [ES]
- k3  [ES]
Similarly for other species...
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System of Simple, Simultaneous Equations
HOW DYNAFIT PROCESSES YOUR BIOCHEMICAL EQUATIONS
k1
E.S
E+S
k3
"The LEGO method"
E+P
k2
of deriving rate equations
Rate terms:
Input (plain text file):
E + S ---> ES
:
k1
k1  [E]  [S]
ES ---> E + S
:
k2
k2  [ES]
ES ---> E + P
:
k3
k3  [ES]
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Rate equations:
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DynaFit can analyze many types of experiments
MASS ACTION LAW AND MASS CONSERVATION LAW IS APPLIED IN THE SAME WAY
biophysics
enzymology
chemistry
EXPERIMENT
Kinetics (time-course)
DYNAFIT DERIVES A SYSTEM OF ...
Ordinary differential equations (ODE)
Equilibrium binding
Nonlinear algebraic equations
Initial reaction rates
Nonlinear algebraic equations
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Example: DNA + clamp / clamp loader complex
DETERMINE ASSOCIATION AND DISSOCIATION RATE CONSTANT IN AN A + B  AB SYSTEM
Typical email from a Ph.D. student:
Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab)
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Example: Experimental setup
ALL COMPONENTS PRESENT AT EQUAL CONCENTRATIONS
1.
2.
3.
pre-mix sliding clamp (C) + clamp loader (L) to form C.L complex;
add DNA solution;
observe the formation of C.L.DNA ternary complex over time
final concentrations:
100 nM clamp
100 nM loader
100 nM DNA
... gp45 labeled with Cy5 acceptor dye
... gp44/62
... primer labeled with Cy3 donor dye
C.L complex has estimated Kd = 1 nM, so
C.L  C+L dissociation upon adding DNA should be negligible
Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab)
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Example: Raw data
JUST BECAUSE THE DATA FIT TO A MODEL DOES NOT MEAN THAT THE MODEL IS CORRECT!
raw fluorescence F fit to
fluorescence
F = A0 + A1 exp (-k t)
exponential model
fits the data well
but it is theoretically
invalid!
time
Courtesy of Senthil Perumal, Penn State University (Steven Benkovic lab)
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Example: Anatomy of DynaFit scripts
DYNAFIT SOFTWARE IS DRIVEN BY TEXT “SCRIPTS” - MINIATURE “COMPUTER PROGRAMS”
keywords
sections
arbitrary names
initial estimates
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Example: DynaFit tutorial
YOUR FIRST DYNAFIT DATA-ANALYSIS SESSION
1.
Start DynaFit
2.
Select menu “File ... Open” or press Ctrl+O
3.
Navigate to file
TUTORIAL
./courses/bkeb/lec-1/a+b/fit-001.txt
4.
Select menu “File ... Try” or press Ctrl+T
This is the initial estimate
5.
Select menu “File ... Run” or press Ctrl+U
Wait several seconds to finish the analysis
6.
Select menu “View ... Results in External Browser”
Navigate in the output files
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Example: Detailed explanation
A BIT OF THEORY
1.
Reaction order
2.
Units and dimensions (scaling)
3.
The DynaFit model for biomolecular kinetics
4.
Initial estimates of model parameters
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Molecularity and reaction order
IN PRACTICE WE ENCOUNTER ONLY ZERO-, FIRST-, AND SECOND-ORDER REACTIONS
ORDER
zero-
firstuni-molecular
monomolecular
secondbimolecular
PHYSICAL
MEANING
DYNAFIT
NOTATION
NOTATION
constant-rate
influx or efflux
isomerization or
dissociation of
one molecule
binding
(association) of
two molecules
A
A
k1
k’1
A+B
B
B+C
k2
C
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X -->
:
v
A --> B
C --> A + B
:
:
k1
k1’
A + B --> C
:
k2
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Reversible reactions and reaction mechanisms
DYNAFIT CAN HANDLE AN ARBITRARY NUMBER OF ELEMENTARY REACTIONS IN A MECHANISM
REVERSIBLE REACTION
A+B
k1
k2
DYNAFIT
NOTATION
top
A + B <==> C
C
:
k1
k2
left
MULTI-STEP MECHANISM
DYNAFIT
NOTATION
+B k
1
A
+C
k3
AC
AB
k4
k2
k5
X
A + B <==> AB
:
k1
k2
A + C <==> AC
:
k3
k4
AC ---> X
:
k5
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Dimensions of rate constants
CAREFUL ABOUT DIMENSIONS OF RATE CONSTANTS! DIMENSIONAL ANALYSIS
A+B
k1
k2
forward and reverse
reaction rates:
AB
quantity
v
v = k1 [A] [B]
[X]
v = k2 [AB]
k1 , k 2
dimension
concentration / time
concentration
?
EXAMPLE
dimensional analysis of k1 (bimolecular association rate constant)
v = k1 [A] [B]
k1 =
v
{conc.} / {time}
1
[A] [B]
{conc.}  {conc.}
=
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{conc.}  {time}
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Dimensions of rate and equilibrium constants
SUMMARY
A+B
A
k1
k2
k3
k4
dissociation rate constant
k2
1/sec
association rate constant
k1
1/(M sec)
dissociation equilibrium constant
Kd = k2 /k1
M
association equilibrium constant
Ka = k1 /k2
1/M
AB
A’
 rate constant
k3
1/sec
 rate constant
k4
1/sec
 equilibrium constant
K = k3 /k4
--
 equilibrium constant
K = k4 /k3
--
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Example: Units (scaling) of rate constants
ALL UNITS ARE ARBITRARY BUT MUST BE IDENTICAL THROUGHOUT THE ENTIRE SCRIPT!
[mechanism]
DNA + Clamp.Loader <==> Complex
[constants]
kon = 1 ?
koff = 1 ?
kon
koff
= 1 µM-1s-1 = 106 M-1.s-1
[concentrations]
DNA
= 0.1
Clamp.Loader = 0.1
concentration units of rate constants: µM
time units of rate constants: sec
100 nM
[responses]
Complex = 1 ?
[data]
file
offset
:
./.../d1-edit.txt
auto ?
time, s
signal, eV
0.52
0.56
0.60
0.64
0.68
0.72
...
0.3526
0.3484
0.3485
0.3454
0.3499
0.3502
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Example: The “response” coefficient
MOLAR “RESPONSE” = PROPORTIONALITY FACTOR LINKING CONCENTRATIONS TO SIGNAL
[mechanism]
DNA + Clamp.Loader <==> Complex
:
kon
koff
[constants]
kon = 1 ?
koff = 1 ?
one concentration unit (in this case 1 µM)
of Complex will produce an increase in the
signal equal to 1.00 instrument units
[concentrations]
DNA
= 0.1
Clamp.Loader = 0.1
[responses]
Complex = 1.00 ?
[data]
file
offset
./.../d1-edit.txt
auto ?
time, s
signal, eV
0.52
0.56
0.60
0.64
0.68
0.72
...
0.3526
0.3484
0.3485
0.3454
0.3499
0.3502
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Example: Initial estimates
NONLINEAR REGRESSION ANALYSIS ALWAYS REQUIRES INITIAL ESTIMATES OF THE SOLUTION
the initial estimate of rate constants
[mechanism]
DNA + Clamp.Loader <==> Complex
[constants]
kon = 1
koff = 1
?
?
[data]
file
offset
kon
koff
optimized model parameters
[concentrations]
DNA
= 0.1
Clamp.Loader = 0.1
[responses]
Complex = 1
:
A VERY DIFFICULT PROBLEM:
?
./.../d1-edit.txt
auto ?
HOW TO GUESS
“GOOD ENOUGH”
INITIAL ESTIMATES
OF RATE CONSTANTS?
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Example: “Good” initial estimate
data
model
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Example: “Good” initial estimate – results
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Example: “Bad” initial estimate
model
a hundred times
smaller / larger
data
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Example: “Bad” initial estimate – results
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initial
estimate
sum of
squares
“good”
k1 = 1
k2 = 1
0.002308
1.00
k1 = 2.2 ± 0.5
k2 = 0.030 ± 0.015
“bad”
Example: “Good” vs. “Bad” results - comparison
relative
“best-fit”
sum of sq. constants
k1 = 100
k2 = 0.01
0.002354
1.02
k1 = 0.2 ± 3.4
k2 = 0.2 ± 0.6
Kd, nM
= k2/k1
13 nM
1000 nM
DynaFit warnings from running the “bad” estimate:
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Example: “Good” vs. “Bad” results - comparison
From “good” initial estimate
From “bad” initial estimate
not very encouraging!
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Example: “Good” vs. “Bad” results - summary
1.
Initial estimates “off” by a factor of 100 can produce misleading results.
2.
The data/model overlay may “look good”, but the results may be invalid.
3.
The same applies to the residual sum of squares (only 2% difference).
4.
The only indication that something went wrong might be:
a.
b.
5.
huge standard errors of model parameters; and
various warnings from the least-squares fitter
The simplest possible safeguard: Use several different initial estimates?
Disadvantage: how do we know which multiple estimates?
LOOKING AHEAD
DynaFit offers more reliable and convenient safeguards
• Global minimization
• Systematic combinatorial scan
• Confidence interval search
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Summary and conclusions
NUMERICAL MODELS IN BIOCHEMISTRY AND BIOPHYSICS: BETTER THAN ALGEBRAIC EQUATIONS
1.
Numerical models are applicable to all experimental conditions.
2.
Numerical models apply uniformly to all types of experiments:
No more “large excess of this over that”.
a.
b.
c.
reaction progress (kinetics);
equilibrium composition (binding);
enzyme catalysis.
3.
Numerical models can be automatically derived by computer.
4.
Main disadvantage: requirement for specialized software.
5.
Not a “silver bullet”! Example: the initial estimate problem.
No more looking up algebraic equations – if they exist at all.
But DynaFit is free to academic users.
But this is not specific to numerical models (applies to algebraic models, too).
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